My Math Forum I can't understand this proof about Uniform Structures

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 March 12th, 2010, 09:34 AM #1 Newbie   Joined: Mar 2010 Posts: 7 Thanks: 0 I can't understand this proof about Uniform Structures There is a uniform structure U, defining a uniform topology associated with U. For every U $\in$ U, U[x] is U evaluated at x, and U[A] is the union of U evaluated at every point in A. The theorem is: U[A] $\subseteq$ B for some U, then $\stackrel{-}{A}\subseteq B$ The proof is below: We may suppose that U is symmetric since there is a symmetric V such that V[A] $V[A]\subseteq (V \circ V)[A] \subseteq U[A]$ (I don't understand how we can conclude that U is symmetric based on the fact that it has a symmetric subset; I understand the rest of the proof.) $x\in\stackrel{-}{A} \Rightarrow \exists a\in A: a\in U[x]$ $\Rightarrow (x,a) \in U \Rightarrow (a,x) \in U$ (by symmetry of U) $\Rightarrow x \in U[a] \subseteq U[a] \subseteq U[A] \subseteq B$ (implies closure of A is in B) I figured this out, instead of U, use the arguments on the symmetric V. V[a] is a subset of U[a].

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